Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 28 + \left(22 a + 5\right)\cdot 103 + \left(42 a + 60\right)\cdot 103^{2} + \left(13 a + 62\right)\cdot 103^{3} + \left(17 a + 70\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 7\cdot 103 + 16\cdot 103^{2} + 44\cdot 103^{3} + 46\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 21\cdot 103 + 32\cdot 103^{2} + 43\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 82 a + 49 + \left(80 a + 6\right)\cdot 103 + \left(60 a + 80\right)\cdot 103^{2} + \left(89 a + 33\right)\cdot 103^{3} + \left(85 a + 74\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 36\cdot 103 + 56\cdot 103^{2} + 11\cdot 103^{3} + 73\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 a + 73 + \left(68 a + 16\right)\cdot 103 + \left(42 a + 45\right)\cdot 103^{2} + \left(20 a + 16\right)\cdot 103^{3} + \left(58 a + 25\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 a + 47 + \left(34 a + 8\right)\cdot 103 + \left(60 a + 19\right)\cdot 103^{2} + \left(82 a + 97\right)\cdot 103^{3} + \left(44 a + 62\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(3,4,5,6,7)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $10$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $70$ | $3$ | $(1,2,3)$ | $1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $360$ | $7$ | $(1,2,3,4,5,6,7)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $360$ | $7$ | $(1,3,4,5,6,7,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
